Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

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8 MSS A Bb B C C D Eb E F F G G Pythagore 141.572 0. 113.69 203.91 317.6 407.82 521.51 611.73 701.96 815.64 905.87 1019.6 1109.8 Zarlino 47.8701 0. 111.73 203.91 315.64 386.31 498.04 568.72 701.96 813.69 884.36 1017.6 1088.3 Kirnberger2 146.869 0. 90.225 203.91 294.13 386.31 498.04 590.22 701.96 792.18 895.15 996.09 1088.3 Kirnberger3 164.332 0. 90.379 195.19 294.01 386.45 498.05 590.3 698.22 792.33 890.27 995.97 1088.2 Neidhardt 31.6585 0. 113.69 203.91 317.6 384.36 474.58 611.73 748.88 815.64 882.4 1062.9 1066.5 Werkmeister1 180.602 0. 90.225 192.18 294.13 390.22 498.04 588.27 696.09 792.18 888.27 996.09 1092.2 Werkmeister2 119.663 0. 82.405 196.09 294.13 392.18 498.04 588.27 694.13 784.36 890.22 1003.9 1086.3 Werkmeister3 234.585 0. 96.09 203.91 300. 396.09 503.91 600. 701.96 792.18 900. 1002. 1098. Werkmeister4 268.489 0. 90.661 196.2 298.07 395.17 498.04 594.92 697.54 792.62 893.21 1000. 1097.1 Werkmeister5 27.5673 0. 43.305 133.53 176.83 337.44 451.12 470.97 561.19 745.26 835.48 878.79 969.01 meanTone15 80.1584 0 113.69 194.53 308.21 389 502.74 616.4 697.26 810.95 891.79 1005.5 1086.3 meanTone16 117.405 0 109.78 196.09 305.87 392.18 501.96 611.73 698.04 807.2 894.13 1003.9 1090.2 Vallotti 164.255 0 94.13 196.09 298.04 392.18 501.96 592.18 698.04 796.09 894.13 1000 1090.2 BachLehman 266.823 0 103.91 200 305.87 403.91 501.96 603.91 698.04 807.2 901.96 1003.9 1103.9 Lindley 45.6815 0. 125.81 187.88 289.44 379.67 493.35 583.58 685.92 827.76 893.74 1003.5 1121.9 LindleyBis 76.1371 0. 113.69 193.74 296.48 386.71 500.39 590.62 692.18 813.29 903.52 1027.4 1117.6 Lindley94 61.4739 0. 113.69 173.41 273.02 363.25 476.93 567.16 671.46 801.56 891.79 1005.5 1109.8 Sparschuh 47.5277 0. 121.51 189.83 285.53 394.13 495.31 635.19 725.42 839.1 909.78 1016.8 1116. Lehman94 50.7568 0. 135.19 182.4 296.09 386.31 500. 590.22 723.46 837.15 927.37 1019.6 1107.8 Figure 5. Values of MSS for different tunings Zarlino and the meantone temperaments. Remember that MSS measures a closeness between major scales, not the quality of, for instance, the twelve fifths. 2.2. Minor scales. As watchful readers will have noticed, the MSS has so far left aside half of WTC: J.S. Bach also gave his preludes and fugas in all minor tonalities. MSS is originally a measure of ‘diatonicity’: could it also be used for testing minor scales ? If by ‘minor’, one should understand the harmonic minor scale, then the theoretical relevance is poor. 14 But this natural assumption, taught to every kid in music school, has long been challenged by specialists. 15 Perhaps the most ‘natural’ form of the minor is also diatonic, meaning that MSS is relevant both for minor and major tonalities. It might be interesting though to try this comparison on other familiar scales, like the ascending melodic minor [023579a]. 2.3. First five fifths equal. Permutations of the values of the different fifths of LT can be tried. The result is illuminating. As we have already discussed, an optimal value of MSS (= ∞) is attainable, in equal temperament. This is not realistic, as most musicologists now agree that equal temperament was as abhorrent to Bach as it was to many, before and afterwards. So it should come as no surprise that some possible (non equal) temperaments improve the value of MSS. It is the case for Werckmeister IV, which, though rational-valued, is a close approximation to equal temperament. 16 I found several such tunings in trying up all permutations of the values of fifths that Lehman attributed to the three different kinds of loops in Bach’s scribble. For instance the one which corresponds to the sequence of adjustments 0, −2, −2, −2, 1, −2, 1, −1, 0, −1, 0, −2 17 , has a much higher value of MSS. Such artefacts are generally very close to equal temperament. I did not recognize any known tuning in them, though learned readers might. But strikingly again, LT is still a winner when these permutations are confined to the six last fifths, i.e. when we keep the first five fifths equal: here I am discussing adjustments of the form −2, −2, −2, −2, −2 followed by any permutation of 0, 0, 0, −1, −1, −1. This must be relevant, as it was and still is common practice to start the tuning of a keyboard instrument by five equal fifths (often slightly shorter than a pure fifth), thus tuning equally the Guidonian hexachord FCGDAE. I certainly did, long before I ever heard of Bradley Lehman. This is true of all meantone tunings for instance, and also very obvious for a musician accustomed to tonal music, 14 Still I felt compelled to compute the mSS, the equivalent of MSS for harmonic minor scales, and rather strangely, it concurs with Table 5. 15 ‘It may be of some interest that Johann Sebastian Bach in his manual on thorough bass (reprinted in Phillip Spitta’s biography of Bach as Appendix XII to Vol. VII) expressly states the identity of the minor mode with the Aeolian system...” (p. 46, Heinrich Schenker [Harmony], Oswald Jonas, ed., Elisabeth Mann Borghese, trans. 1954, reprinted MIT Press 1973). 16 Hearing WTC in this tuning should be an interesting experience, considering it is so far the only competitor to LT for MSS. 17 Whereas LT gives −2, −2, −2, −2, −2, 0, 0, 0, −1, −1, −1[, 0]. The unit chosen by Lehman is a twelfth of a pythagorean comma.
since it is still desirable to have many equal fifths inside the scale(s) he or she plays in. Many harpsichord, organ, or piano-forte playing readers will acknowledge that they begin their own tunings by equal fifths between F-C-G-D-A-E. See below a discussion of O’Donnell’s objection, though [10]. 2.4. Other values of the wiggles. One weak point of Lehman’s proposition is his arbitrary calibration of the wiggles as multiples of PC/12 (PC = a pythagorean comma). On the other hand, changing this value ever so slightly falls within the approximation involved in any practical tuning – this is not exact science ! On investigation, I tried variants of LT, replacing the unit PC/12, or 2.346 cents, by values close to it. The value of MSS can thus be increased slightly: if the fundamental wiggle is reduced to 2.16 cents, MSS reaches 278.7 instead of 266. This is the best possible improvement in that direction, with T = {0, 95.685, 196.71, 297.795, 393.42, 498.105, 593.73, 698.355, 797.64, 895.065, 997.95, 1091.78} Of course, a difference of 0.186 cents is hardly perceptible by ear – it is roughly a 300th of a comma ! It is only the different order of fifths, the geometry of the scale, and not the change of comma, that explains the record value of MSS; the different value of the comma in Lehman’s first proposition in 1994 (labeled Lehman94 in fig. 5) has nothing to do with its poor value of MSS. 2.5. Other interpretations of the wiggles. Some have questioned the original note of the tuning: is it F – a common practice – or do some further curlicues along the scribble (look again at fig 1) indicate the position of C, or D, or some other note ? As far as the MSS is concerned, this is irrelevant, as a ‘well tempered’ instrument will remain so even if the tuning is transposed, in the sense that all tonalities still sound well. Of course it would be nice to know exactly Bach’s tuning (giving specific different colours, i.e. Affekt, to different tonalities) but MSS cannot discriminate between tunings there, as it is invariant under a change of origin of the tuning. 18 Similarly, MSS cannot help us decide whether turning around the page (as Lehman did) is permitted, or mandatory: the inversion of a tuning shares the same geometric values 19 , and the sequence of wiggles (-1,- 1,-1,0,0,0,-2,-2,-2,-2,-2) has the same MSS as LT – a nice feature for counterpointists. In other words, using Lehman’s recipe with fourths instead of fifths yields exactly the same value of MSS. MSS can help, though, to discuss whether the wiggles mean consecutive fifths or consecutive semitones. This is a strong argument, as O’Donnell points out ([10]) that the order of tonalities in WTC is given in semitones not in fifths – but Chopin, for instance, did contrariwise in his Preludes, which are clearly referring to WTC. I am not sure that I understood O’Donnell rightly, as this sequence of five short successive semitones yields one very poor fourth interval (or a very large fifth); but I computed MSS for those tunings given in Lehman’s comprehensive answer to O’Donnell, in Table 6: MSS A Bb B C C D Eb E F F G G BachLehman 266.823 0 103.91 200 305.87 403.91 501.96 603.91 698.04 807.2 901.96 1003.9 1103.9 O'Donnell 40.0079 0. 137.15 227.37 294.13 384.36 498.04 635.19 678.49 839.1 905.87 1019.6 1109.8 Neidhardt4 110.593 0. 43.305 180.45 247.21 337.44 451.12 588.27 631.57 792.18 858.94 972.63 1062.9 Neidhardt 1732 53.9361 0. 137.15 227.37 294.13 384.36 498.04 588.27 701.96 815.64 929.33 1043. 1109.8 Sorge 54.7281 0. 113.69 227.37 294.13 384.36 498.04 635.19 678.49 815.64 905.87 1043. 1133.2 Figure 6. MSS for O’Donnell-like tunings Again MSS is much greater for LT than for all other tunings. Several falsifications, in the sense of Popper, have been tried. In each case, LT emerges a winner. It is clearly a very special temperament, not only in its closeness to Equal Temperament, but in the homogeneity of all 24 scales. This constitutes a strong vindication of Lehman’s claim. 3. Conclusion and perspectives The MSS criterion arguably puts emphasis on the sameness of fifths inside the temperament. This goes some way towards explaining why permutations of the wiggles fail to improve MSS, when the first five are kept identical (see subsection 2.3); it also explains the fair value of MSS for Pythagorean (Ptolemaic) tuning, for instance. But this argument has limits, as the discussion on all permutations of the wiggles proves. Still is is quite important musically, as keyboard intrument players will readily agree, to find in any major scale a 18 Transposition, in the musical sense, does not change the absolute value of the DFT. 19 The DFT is turned into its conjugate. See [1]. 9
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Thèse de Doctorat de l'Université
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Résumé : Cette thèse sur travaux
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Table des matières Introduction 6
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kis. De ce fait, j'ai tout naturell
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fois ce problème de combinatoire r
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de Quinn 12, qui, pour la première
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OpenMusic permettant, entre autres
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Le premier article offre une synth
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A = {0,1,2,4,5,6}, i.e. A(X) = (1 +
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✦ Fabriquer par l'algorithme de C
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(ou une gamme) remonte à David Lew
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Mon article du Journal of Mathemati
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et de considérer les DFT des appli
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Pour rendre compte de cette dispari
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limitées 54, les pavages, divers t
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OpenMusic réalise les transformati
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ment versé en mathématiques, pour
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Mémoire de thèse Amiot 38 Un sign
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Mémoire de thèse Amiot 40 Or l' o
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Mémoire de thèse Amiot 42 Remerci
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Mémoire de thèse Amiot 44 Je tien
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Mémoire de thèse Amiot 46 ✦ Ami
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Annexe I Articles figurant dans le
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du pattern, qui va donc commencer p
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Un musicien attend pour rentrer ICI
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ainsi que leurs produits. Comme ce
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par l’automorphisme de FROBENIUS
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PROPOSITION. ≀ Soit A(X) un polyn
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FIG. 12 - Orbite d’un motif sous
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. Un domaine K (compact d’intéri
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période 900 : A = (0, 36, 72, 100,
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Journal of Mathematics and Music Vo
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alternative title 3 Lemma 1.2 The F
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Page 94 and 95: 4 autosimilar melodies Example 1.3
Page 96 and 97: 6 autosimilar melodies Proposition
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Page 100 and 101: 10 autosimilar melodies homothety h
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Page 106 and 107: 16 autosimilar melodies 5.2 Example
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Page 110 and 111: 20 autosimilar melodies [8] Fripert
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Page 114 and 115: 24 autosimilar melodies But in Z15
Page 116 and 117: 26 autosimilar melodies At this jun
Page 118 and 119: 2 circle of fifths begins with five
Page 120 and 121: 4 (it is the Fourier Transform of t
Page 122 and 123: 6 Definition 2. A temperament, or t
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Page 128 and 129: 12 i.e. an arithmetic progression w
Page 130 and 131: 2 EMMANUEL AMIOT If we visualize Zn
Page 132: 4 EMMANUEL AMIOT Theorem 3. (1) Til
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Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

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